Coupled Oscillator: horizontal Mass-Spring

Problem Statement

Write up on the Wiki a solution of a coupled oscillator problem like the coupled pendulum. Use State Space methods. Describe the eigenmodes and eigenvectors of the system.

Initial Conditions:

$m_{1}=10kg\,$

$m_{2}=10kg\,$

$k1=25N/m\,$

$k2=75N/m\,$

$k3=50N/m\,$

Equations for M_1

${\begin{alignedat}{3}F&=ma\\F&=m{\ddot {x}}\\-k_{1}x_{1}-k_{2}(x_{1}x_{2})&=m_{1}{\ddot {x_{1}}}\\-{k_{1}x_{1} \over {m_{1}}}-{k_{2}(x_{1}-x_{2}) \over {m_{1}}}&=m_{1}{\ddot {x_{1}}}\\-{k_{1}x_{1} \over {m_{1}}}-{k_{2}(x_{1}-x_{2}) \over {m_{1}}}&={\ddot {x_{1}}}\\-{k_{1}+k_{2} \over {m_{1}}}x_{1}+{k_{2} \over {m_{1}}}x_{2}&={\ddot {x_{1}}}\\\end{alignedat}}$

Equations for M_2

${\begin{alignedat}{3}F&=ma\\F&=m{\ddot {x}}\\-k_{2}(x_{2}-x_{1})&=m_{2}{\ddot {x_{2}}}\\{-k_{2}(x_{2}-x_{1}) \over {m_{2}}}&={\ddot {x_{2}}}\\-{k_{2} \over {m_{2}}}x_{2}+{k_{2} \over {m_{2}}}x_{1}&={\ddot {x_{2}}}\\\end{alignedat}}$

Additional Equations

${\dot {x_{1}}}={\dot {x_{1}}}$

${\dot {x_{2}}}={\dot {x_{2}}}$

State Equations

${\begin{bmatrix}{\dot {x_{1}}}\\{\ddot {x_{1}}}\\{\dot {x_{2}}}\\{\ddot {x_{2}}}\end{bmatrix}}\,$ = ${\begin{bmatrix}0&1&0&0\\{\frac {(k_{1}-k_{2})}{m_{1}}}&0&{\frac {-k_{1}}{m_{1}}}&0\\0&0&0&1\\{\frac {k_{1}}{m_{2}}}&0&{\frac {(k_{1}+k_{2})}{m_{2}}}&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\{\dot {x}}_{1}\\x_{2}\\{\dot {x}}_{2}\end{bmatrix}}+{\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}}{\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

With the numbers...

${\begin{bmatrix}{\dot {x_{1}}}\\{\ddot {x_{1}}}\\{\dot {x_{2}}}\\{\ddot {x_{2}}}\end{bmatrix}}\,$ = ${\begin{bmatrix}0&1&0&0\\{\frac {(-50N/m)}{10kg}}&0&{\frac {-25N/m}{10kg}}&0\\0&0&0&1\\{\frac {25N/m}{10kg}}&0&{\frac {(100N/m)}{10kg}}&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\{\dot {x}}_{1}\\x_{2}\\{\dot {x}}_{2}\end{bmatrix}}$

Eigen Values

Once you have your equations of equilibrium in matrix form you can plug them into a calculator or a computer program that will give you the eigen values automatically. This saves you a lot of hand work. Here's what you should come up with for this particular problem given these initial conditions.

Given

$m_{1}=10kg\,$

$m_{2}=10kg\,$

$k_{1}=25\,{N \over {m}}$

$k_{2}=50\,{N \over {m}}$

We now have

${\begin{bmatrix}{\dot {x_{1}}}\\{\ddot {x_{1}}}\\{\dot {x_{2}}}\\{\ddot {x_{2}}}\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\-5&0&-2.5&0\\0&0&0&1\\2.5&0&10&0\end{bmatrix}}{\begin{bmatrix}{x_{1}}\\{\dot {x_{1}}}\\{x_{2}}\\{\dot {x_{2}}}\end{bmatrix}}+{\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

From this we get

$\lambda _{1}=\,$

$\lambda _{2}=\,$

$\lambda _{3}=\,$

$\lambda _{4}=\,$

Eigen Vectors

Using the equation above and the same given conditions we can plug everything to a calculator or computer program like MATLAB and get the eigen vectors which we will denote as $k_{1},k_{2},k_{3},k_{4}\,$.

$k_{1}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

$k_{2}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

$k_{3}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

$k_{4}={\begin{bmatrix}0\\0\\0\\0\end{bmatrix}}$

Matrix Exponential

In this section we will use matrix exponentials to solve the same problem. First we start with this identity.

$z=Tx\,$

This can be rearranged by multiplying the inverse of T to the left side of the equation.

$T^{-1}z=x\,$

Now we can use another identity that we already know

${\dot {x}}=Ax$

Combining the two equations we then get

$T^{-1}{\dot {z}}=AT^{-1}z$

Multiplying both sides of the equation on the left by T we get

${\dot {z}}=TAT^{-1}z$

We also know what T equals and we can solve it for our case

$T^{-1}=[k_{1}|k_{2}|k_{3}|k_{4}]\,$

$T^{-1}={\begin{bmatrix}0.2149i&-0.2149i&-0.3500i&0.3500i\\-0.5722&-0.5722&0.4157&0.4157\\-0.2783i&0.2783i&-0.5407i&0.5407i\\0.7409&0.7409&0.6421&0.6421\end{bmatrix}}$

Taking the inverse of this we can solve for T

$T={\begin{bmatrix}-1.2657i&-0.4753&0.8193i&0.3077\\1.2657i&-0.4753&-0.8193i&0.3077\\0.6514i&0.5484&0.5031i&0.4236\\-0.6514i&0.5484&-0.5031&0.4236\end{bmatrix}}$